Cosmology Rs Cosmo Module 002

What the module is

Cosmology RS Module 002 is a short file in the Recognition Science framework's machine-checked library of formal theorems. The file defines a cost function, proves three elementary facts about it, and attaches a research note about the cosmological constant. The note is not a theorem; the proofs do not touch cosmology.

The cost function is cost, a measure of how expensive a recognition event is when an observed quantity m is compared to an expected quantity e. The module defines it as J(m/e), where J(x) = (x + 1/x)/2 - 1. The three proved facts are: the cost is zero when m equals e, the cost is never negative for positive inputs, and the golden-ratio constant phi is greater than 1.5. These are general properties of J, not claims about any physical system.

The research note says the product of the cosmological constant and the Planck length squared, Lambda ell_P^2, equals 8*phi^5/45, about 1.99e-122, close to the measured value of 1.99e-122. The note labels this as a research note, not a result. The module proves none of it. To become a theorem about cosmology, the module would need to define m and e in cosmological terms, which it does not do.

In plain language: the module proves three small, true facts about a cost formula, and it records an idea about the cosmological constant that remains unproved. The idea is a hypothesis, not an established result.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove the cosmological constant value. The module does not define m or e in cosmological terms. The module does not establish any physical prediction about the universe.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/RS_Cosmo_Module_002.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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